Picture Fuzzy Choquet Integral Based Geometric Aggregation Operators and Its Application to Multi Attribute Decision-Making

Multi attribute decision-making (MADM) is the process of selecting the most suitable alternative from among a number of available alternatives. Zadeh [1] introduced the main concept of fuzzy set (FS) theory in 1965 and examined the degree of membership of an object in a set, which ranges from zero to one. Later, Intuitionistic fuzzy set (IFS) was established by Atanassov [2] in 1986 as an extension of FS by including the degree of non-membership as well as the degree of membership to each object in a set and demonstrating various properties associated with operations and relations over sets. If an expert obtains an opinion from a certain person regarding any object, that person may indicate that 0.5 is the likelihood that the statement is true, 0.3 is the possibility that the statement is false and 0.2 is that he or she is not sure. As a result, the role of neutrality is not handled by FSs or IFSs but must be addressed in real-life decision making. Cuong [3, 4] proposed a picture fuzzy set (PFS) and investigated certain essential operations and properties. The PFS is influenced by the degree of membership, the degree of neutral membership and the degree of non-membership. The only requirement is that the sum of the three degrees be less than or equal to one. Essentially, PFS based models may be applied to situations requiring human opinions involving more sorts of answers: yes, abstain, no, refuse, which cannot be adequately described in FS and IFS. Dutta et al. [5] defined (α, δ, β)-cuts under picture fuzzy (PF) environment. Many researchers introduced PFSs into the MADM field to express fuzzy evaluations due to their high performance in dealing fuzzy information.

Aggregation operators (AOs) are especially important in the decision-making (DM) process since they combine all of the provided individual assessment values into a single form. Yager [6] explored the features of an ordered weighted AOs and described a new form of AOs. Cuong et al. [7] investigated picture t-norm and picture t-conorms and its properties for PFSs. Wei [8] described various PF Weighted average (WA) and Weighted geometric (WG) operators as well as PF hybrid aggregators and their applications. Wang et al. [9] studied MADM problems utilising some PF geometric operators. Garg [10] demonstrated a series of AOs for the PFSs that were used to solve the multi criteria decision-making problem. Wei [11] used arithmetic and geometric AOs, Hamacher operations with PF environment to study the MADM problem. Jana et al. [12] developed PF Dombi arithmetic and geometric AOs are used to solve MADM problem. Zhang et al. [13] developed new PF operational laws based on Dombi t-norm and t-conorm and utilised Heronian mean (HM) information AO to combine picture fuzzy numbers (PFN) and the suggested operators not only combine individual attribute values but they are also able to model the general correlation between attributes. Wang et al. [14] used a case study regarding financial investment risk to provide various picture fuzzy aggregation operators (PFAO) that were based on the classic Muirhead mean (MM) operators and developed the MADM approach. Xu et al. [15] proposed a family of MM operators with PF environment and its properties are investigated. Jana et al. [16] developed MADM approaches for enterprise performance evaluation using PF Hamacher AOs. Khan et al. [17] suggested an AOs based on PF Einstein operations for the MADM problem using PF environment. Qiyas et al. [18] developed some AO based on the idea of yager operators with PF environment based on Archimedean t-norm and t-conorm. Qin et al. [19] proposed a novel MADM approach that uses a set of Archimedean power Maclaurin symmetric mean (MSM) operators of PFNs. Tapan Senapati [20] introduced the AOs of PFNs and a few AOs mainly PF Aczel-Alsine average AOs as well as these operators to develop a method for solving MADM in a PF environment. Table 1 is a list of the AOs of PFNs on which these approaches are based.

The above mentioned PFAOs are initiated within a presumption: The criteria are independent. In most of the MADM problems which are dependent on the contrary, they are correlative. In such cases, the important degrees of criteria are given as fuzzy measures (FM) rather than weights, because weighted AOs cannot be aggregated. FMs can well reflect the correlative relationships between criteria sets, such as redundant, independent, and complimentary. In reality, FMs are extensions of weights due to the fact that the sum of all FMs of criteria set can exceed one, whereas the FM of the entire set_is_restricted to one. Consequently, the FMs are more flexible. Choquet integral is an efficient method for aggregating ranking based on correlated criteria. To handle multi-criteria group decision making (MCGDM) problems where the criteria and expert opinions frequently involve inter-dependent or interaction phenomena among criteria. Tan et al. [21] investigated the generalised IF ordered geometric averaging operator. Zhang et al. [22] developed the Einstein interval IF Choquet geometric operator and investigated the relationship with the IF Choquet geometric operator. Singh et al. [23] described Choquet averaging and geometric mean operators for PFSs and also presented a VIKOR for MCGDM problems based on PF Choquet integrals. IF arithmetic and hybrid arithmetic AOs are proposed by Jia and Wang [24] based on Choquet integral as well as proposed operator computes the correlative criteria and combines the weight of position.

In MADM problems, the decision-making criteria may be dependent or independent. FMs were first presented by Sugeno [25] in 1974 to model interactions between decision criteria. The aforementioned AOs, which are unable to properly handle the specific conditions with correlative criteria and weighted positions, we define an AOs as the Choquet integral picture fuzzy geometric aggregation (CIPFGA) operator. After that, we combine the FMs of the criteria and the weights of the positions to construct the Choquet integral picture fuzzy hybrid geometric aggregation (CIPFHGA) operator.

The summary of the motivations of this article are outlined as follows:

• When compared to weighted AOs, the CIPFGA operator result can indicate the connections between the criteria.
• In comparison to the Choquet integral based AOs (Singh et al. [23]), the CIPFGA operator result may not only reflect the comprehensive connections of each criteria, but also reflect the unique relevance of each criteria.
• When compared to existing AOs, the CIPFHGA operator results can reflect weights of position. which are listed in Table 2.

The summary of the contributions of this article are outlined as follows:

• It has been demonstrated that the CIPFGA operator can effectively combine PFNs with correlative criteria and it has several properties that have been proved.
• The CIPFHGA operator is described to aggregate PFNs using a combination of FMs and weights position, with various properties are proved.
• To solve decision-making challenges in the PF environment, a MADM approach based on the CIPFHGA operator is provided.

Table 2. A comparison of the existing operators and the proposed operator

From Table 2, existing AOs [8-10, 16-18] are not dealing with FMs and correlative relationships among criteria sets, whereas the proposed CIPFHGA operator completely deals with them.

The present article is structured in a comprehensive manner into seven detailed sections: In Section 2, we briefly review the concepts with respect to the PFSs and the fuzzy measures. In Section 3, the CIPFGA and CIPFHGA operators are defined in detail, along with their properties. Section 4, a MADM approach that relies on the CIPFHGA operator is proposed. Section 5, applying the proposed MADM approach to the numerical problem to demonstrate its practicality. Section 6, Comparative analyses are conducted to demonstrate the advantages of our suggested MADM approach over existing aggregation operators. Conclusions and potential study directions are discussed in the final section.

The Choquet integral based CIPFGA operator is defined in this section and it is combined with the FMs of criteria and the weights of positions to define the CIPFHGA operator.

3.1 CIPFGA operator

Wei [8] and Garg [10] presented certain AOs to aggregate PF environment for MADM problems under the premise of independent criteria. In this part, we propose the Choquet integral operator for the PF environment, which provides for the interaction phenomena between the criteria represented by FM. The CIPFGA operator is defined as follows.

Definition 7: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and ϑ be a family of PFNs and FM on S. The discrete choquet integral CIPFGA operator of α_j with respect to ϑ is define as

$\begin{align}  & CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\alpha }_{1}}\frac{\sum\limits_{{{R}_{1}}\subseteq {{P}_{1}}\left( C \right)}{\vartheta \left( {{R}_{1}}\cup \left\{ S \right\} \right)-\vartheta \left( {{R}_{1}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}}\oplus  \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\alpha }_{2}}\frac{\sum\limits_{{{R}_{2}}\subseteq {{P}_{2}}\left( C \right)}{\vartheta \left( {{R}_{2}}\cup \left\{ {{S}_{2}} \right\} \right)-\vartheta \left( {{R}_{2}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}}\oplus  \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cdots \oplus {{\alpha }_{n}}\frac{\sum\limits_{{{R}_{n}}\subseteq {{P}_{n}}\left( C \right)}{\vartheta \left( {{R}_{n}}\cup \left\{ {{S}_{n}} \right\} \right)-\vartheta \left( {{R}_{n}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}} \\\end{align}$     (3)

where, $R_j \subseteq P_j(S), P_j(S)=P\left(S\left\{S_j\right\}\right)$  Indicates the power of S. For convenience, let ${{\beta }_{j}}=\frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}},$ then Eq. (3) can be simplified as $CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\underset{j=1}{\overset{n}{\mathop{\otimes }}}\,{{\alpha }_{j}}^{{{\beta }_{j}}}.$

Theorem 1: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and ϑ be a family of PFNs and FM on S. Using the CIPFGA operator, their aggregated value is also a PFN and 

$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},\cdots ,{{\alpha }_{n}} \right)\,=\left\langle \begin{align}  & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$          (4)

where, ${{\beta }_{j}}=\frac{\sum\limits_{{{B}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}},$ $R_j \subseteq P_j(S), P_j(S)=P\left(S\left\{S_i\right\}\right)$.

Proof

From the definition (4), the first results come quickly Below, we use mathematical induction on n to show that Eq. (4).

When n=2, since:

$\alpha _{1}^{{{\beta }_{1}}}=\left\langle {{\left( {{\mu }_{{{\alpha }_{1}}}}+{{\eta }_{{{\alpha }_{1}}}} \right)}^{{{\beta }_{1}}}}-\eta _{{{\alpha }_{1}}}^{{{\beta }_{1}}},\eta _{{{\alpha }_{1}}}^{{{\beta }_{1}}},1-{{\left( 1-{{\nu }_{{{\alpha }_{1}}}} \right)}^{{{\beta }_{1}}}} \right\rangle $

$\alpha _{2}^{{{\beta }_{2}}}=\left\langle {{\left( {{\mu }_{{{\alpha }_{2}}}}+{{\eta }_{{{\alpha }_{2}}}} \right)}^{{{\beta }_{2}}}}-\eta _{{{\alpha }_{2}}}^{{{\beta }_{2}}},\eta _{{{\alpha }_{2}}}^{{{\beta }_{2}}},1-{{\left( 1-{{\nu }_{{{\alpha }_{2}}}} \right)}^{{{\beta }_{2}}}} \right\rangle $

Then, $\operatorname{CIPFGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=\alpha_1^{\beta_1} \otimes \alpha_2^{\beta_2}=$$\left\langle\begin{array}{l}\left(\mu_{\alpha_1}+\eta_{\alpha_1}\right)^{\beta_1}\left(\mu_{\alpha_2}+\eta_{\alpha_2}\right)^{\beta_2}-\eta_{\alpha_1}^{\beta_1} \eta_{\alpha_2}^{\beta_2}, \\ \eta_{\alpha_1}^{\beta_1} \eta_{\alpha_2}^{\beta_2}, 1-\left(1-v_{\alpha_1}\right)^{\beta_1}\left(1-v_{\alpha_2}\right)^{\beta_2}\end{array}\right\rangle$.

If Eq. (4) holds for n=k, that is:

$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},\cdots ,{{\alpha }_{k}} \right)\,=\left\langle \begin{align}  & \prod\limits_{j=1}^{k}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{k}{{{(1-{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$

Then, for n=k+1, we have:

$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{k+1}} \right)=\left\langle \begin{align}  & \prod\limits_{j=1}^{k}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{k}{{{(1-{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle \centerdot$

$\,\left\langle \begin{align}  & {{({{\mu }_{{{\alpha }_{k+1}}}}+{{\eta }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}}-{{({{\eta }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}}, \\ & {{({{\eta }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}},1-{{(1-{{\nu }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}} \\\end{align} \right\rangle$

$\,=\left\langle \begin{align}  & \prod\limits_{j=1}^{k+1}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{k+1}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{k+1}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{k+1}{{{(1-{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$

That is to say, the condition n=k+1 is satisfied by Eq. (4).

As a result, Eq. (4) obtains for all n, concluding the proof of Theorem 1.

Theorem 2: (Idempotency) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1$, $2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$. If all $\alpha_j$ are equal, i.e., $\alpha_j=\alpha: \forall \mathrm{j}$, then CIPFGA $\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=\alpha$.

Proof

Since α_j = α = ⟨μ_α, η_α, ν_α⟩ according to theorem 1, we have:

$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\,\left\langle \begin{align}  & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle =\left\langle {{\mu }_{\alpha }},{{\eta }_{\alpha }},{{\nu }_{\alpha }} \right\rangle =\alpha$

where, ${{\beta }_{j}}=\frac{\sum\limits_{{{B}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}},$ $R_j \subseteq P_j(S), P_j(S)=P\left(S\left\{S_j\right\}\right)$.

Theorem 3: (Monotonicity) Let $\alpha_{1 j}=\left(\mu_{\alpha_{1 j}}, \eta_{\alpha_{1 j}}, v_{\alpha_{1 j}}\right)$ and $\alpha_{2 j}=\left(\mu_{\alpha_{2 j}}, \eta_{\alpha_{2 j}}, v_{\alpha_{2 j}}\right)(j=1,2, \ldots, n)$ be two family of PFNs on $S$ and $\vartheta$ be a FM on S. If $\mu_{\alpha_{1 j}} \leq \mu_{\alpha_{2 j}}, \eta_{\alpha_{1 j}} \geq \eta_{\alpha_{2 j}}$ and $v_{\alpha_{1 j}} \geq v_{\alpha_{2 j}^{\prime}} \quad$ then $\operatorname{CIPFAA}\left(\alpha_{11}, \alpha_{12}, \ldots, \alpha_{1 n}\right) \leq$ $\operatorname{CIPFAA}\left(\alpha_{21}, \alpha_{22}, \ldots, \alpha_{2 n}\right)$.

Proof

$\,CIPFGA\left( {{\alpha }_{11}},{{\alpha }_{12}},...,{{\alpha }_{1n}} \right)=\left\langle \begin{align}  & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{1j}}}}+{{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$

$CIPFGA\left( {{\alpha }_{21}},{{\alpha }_{22}},...,{{\alpha }_{2n}} \right)\,=\left\langle \begin{align}  & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{2j}}}}+{{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle $

Since $\mu_{\alpha_{1 j}} \leq \mu_{\alpha_{2 j}}, \eta_{\alpha_{1 j}} \geq \eta_{\alpha_{2 j}}$ and $v_{\alpha_{1 j}} \geq v_{\alpha_{2 j}}$  we have:

$\prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{1j}}}}+{{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}$$\le \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{2j}}}}+{{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}-\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}},$

$\prod\limits_{j=1}^{n}{{{\left( {{\eta }_{{{\alpha }_{1j}}}} \right)}^{{{\beta }_{j}}}}}\ge \prod\limits_{j=1}^{n}{{{\left( {{\eta }_{{{\alpha }_{2j}}}} \right)}^{{{\beta }_{j}}}}}$ and $1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}\ge 1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}.$

Hence

$\prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{1j}}}}+{{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}-\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}-1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}}$

$\le \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{2j}}}}+{{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}},\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}}$

i.e.,

$\operatorname{CIPFGA}\left(\alpha_{11}, \alpha_{12}, \ldots, \alpha_{1 n}\right) \leq \operatorname{CIPFGA}\left(\alpha_{21}, \alpha_{22}, \ldots, \alpha_{2 n}\right)$

Theorem 4: (Boundedness) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, \nu_{\alpha_j}\right\rangle(j=1$ $2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on S.

If $\alpha^{-}=\left\langle\min _j\left(\mu_{\alpha_j}\right), \max _j\left(\eta_{\alpha_j}\right), \max _j\left(v_{\alpha_j}\right)\right\rangle$ and $\alpha^{+}=$ $\left(\max _j\left(\mu_{\alpha_j}\right), \min _j\left(\eta_{\alpha_j}\right), \min _j\left(v_{\alpha_j}\right)\right\rangle \quad$ then $\quad \alpha^{-} \leq$ $\operatorname{CIPFGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right) \leq \alpha^{+}$.

Proof

Using theorem 2, α^- and α⁺ can be defined as CIPFGA(α^-, α^-, ..., α^-) and CIPFGA (α⁺, α⁺, ..., α⁺) respectively.

Since

${{\alpha }^{-}}=\left\langle \underset{j}{\mathop{\min }}\,\left( {{\mu }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\max }}\,\left( {{\eta }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\max }}\,\left( {{\nu }_{{{\alpha }_{j}}}} \right) \right\rangle =\left\langle {{\mu }_{{{\alpha }^{-}}}},{{\eta }_{{{\alpha }^{-}}}},{{\nu }_{{{\alpha }^{-}}}} \right\rangle$

${{\alpha }^{+}}=\left\langle \underset{j}{\mathop{\max }}\,\left( {{\mu }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\min }}\,\left( {{\eta }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\min }}\,\left( {{\nu }_{{{\alpha }_{j}}}} \right) \right\rangle =\left\langle {{\mu }_{{{\alpha }^{+}}}},{{\eta }_{{{\alpha }^{+}}}},{{\nu }_{{{\alpha }^{+}}}} \right\rangle$

Then $\mu_{\alpha^{-}} \leq \mu_{\alpha_j}, \eta_{\alpha^{-}} \geq \eta_{\alpha_j}, v_{\alpha^{-}} \geq v_{\alpha_j}, \mu_{\alpha^{+}} \geq \mu_{\alpha_j}, \eta_{\alpha^{+}} \leq$ $\eta_{\alpha_j}$ and $\nu_{\alpha^{+}} \leq v_{\alpha_j} \forall j$. According to Theorem 3, we have:

$\begin{align}  & CIPFGA\left( {{\alpha }^{-}},{{\alpha }^{-}},...,{{\alpha }^{-}} \right)\le CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\le CIPFGA\left( {{\alpha }^{+}},{{\alpha }^{+}},...,{{\alpha }^{+}} \right). \\\end{align}$

Theorem 5: (Permutation) Let $\alpha_j=\left(\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right)$ and $\alpha_{(j)}=\left(\mu_{\alpha_{(j)}}, \eta_{\alpha_{(j)}}, v_{\alpha_{(j)}}\right)(j=1,2, \ldots, n)$ be two family of PFNs on $S$ and $\vartheta$ be a FM on $S$, where $(j)$ indicates an arbitrary permutation of $j$. then $\operatorname{CIPFGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=$ $\operatorname{CIPFGA}\left(\alpha_{(1)}, \alpha_{(2)}, \ldots, \alpha_{(n)}\right)$.

Proof

$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\left\langle \begin{align}  & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left\langle \begin{align}  & \prod\limits_{\left( j \right)=1}^{n}{{{({{\mu }_{{{\alpha }_{\left( j \right)}}}}+{{\eta }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}}-}\prod\limits_{\left( j \right)=1}^{n}{{{({{\eta }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}}}, \\ & \prod\limits_{\left( j \right)=1}^{n}{{{({{\eta }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}},1-\prod\limits_{\left( j \right)=1}^{n}{{{(1-{{\nu }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}}}} \\\end{align} \right\rangle$

$=CIPFGA\left( {{\alpha }_{\left( 1 \right)}},{{\alpha }_{\left( 2 \right)}},...,{{\alpha }_{\left( n \right)}} \right).$

The proof of Theorem 5 is complete.

Example 1: Let α₁ = ⟨0.4, 0.2, 0.3⟩, α₂ = ⟨0.3, 0.1, 0.5⟩, α₃ = ⟨0.6, 0.2, 0.1⟩ be three PFNs on S, S={S₁, S₂, S₃} and ϑ be a fuzzy measure on S. ϑ(S₁) = 0.2, ϑ(S₂) = 0.3, ϑ(S₃) = 0.4, ϑ(S₁, S₂) = 0.5, ϑ(S₁, S₃) = 0.6, ϑ(S₂, S₃) = 0.7, ϑ(S₁, S₂, S₃) = 1. Then:

$\begin{align}  & CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},{{\alpha }_{3}} \right) \\ & \,\,\,\,\,\,\,=\left\langle 0.4,0.2,0.3 \right\rangle \frac{0.71}{3.09}+\left\langle 0.3,0.1,0.5 \right\rangle \frac{1.03}{3.09}+\left\langle 0.6,0.2,0.1 \right\rangle \frac{1.35}{3.09} \\\end{align}$

$=\left\langle \begin{array}{c} \left((0.4+0.2)^{0.2298}-0.2^{0.2298} \cdot(0.3+0.1)^{0.3333} \\ -0.1^{0.3333} \cdot(0.6+0.2)^{0.4369}-0.2^{0.4369}\right), \\ 0.2^{0.2298} \cdot 0.1^{0.3333} \cdot 0.2^{0.4369}, \\ 1-\left((1-0.3)^{0.2298} \cdot(1-0.5)^{0.3333} \cdot(1-0.1)^{0.4369}\right.\end{array} \right\rangle$

$=\left\langle 0.4356,0.1587,0.3016 \right\rangle .$

3.2 CIPFHGA operator

We propose the CIPFHGA operator, which is motivated by the PFHA operator (Wang et al. [9]).

Definition 8: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and ϑ be a family of PFNs and FM on S and has an weight vector w = (w₁, w₂, ..., w_n)^T with 0≤w_j≤1 and  The (discrete) CIPFHGA operator of α_j with respect to w and ϑ is defined as:

$CIPFHGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\underset{j=1}{\overset{n}{\mathop{\otimes }}}\,\tilde{\alpha }_{(j)}^{{{w}_{j}}}$

where, $\tilde{\alpha}_{(j)}$ is the j^th largest of the $\tilde{\alpha}_j$,

${{\tilde{\alpha }}_{j}}=n\cdot {{\alpha }_{j}}\cdot \frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}},$

If FM ϑ is additive and the FMs of S_j(j=1, 2, ⋯, n) are equivalent, then $R_j \subseteq P_j(S)$, P_j(S)=P (S{S_j}) and n is the balancing coefficient, which functions as a balancing factor in this situation, i.e., reduces to weight $\left(\frac{1}{n}, \frac{1}{n}, \cdots, \frac{1}{n}\right)^T$ then the vector reduces to (α₁, α₂,…, α_n)^T.

$\left( \begin{align}  & n\cdot {{\alpha }_{1}}\frac{\sum\limits_{{{R}_{1}}\subseteq {{P}_{1}}\left( C \right)}{\vartheta \left( {{R}_{1}}\cup \left\{ {{S}_{1}} \right\} \right)-\vartheta \left( {{R}_{1}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}}, \\ & n\cdot {{\alpha }_{2}}\frac{\sum\limits_{{{R}_{2}}\subseteq {{P}_{2}}\left( C \right)}{\vartheta \left( {{R}_{2}}\cup \left\{ {{S}_{2}} \right\} \right)-\vartheta \left( {{R}_{2}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}}, \\ & \cdots ,n\cdot {{\alpha }_{n}}\frac{\sum\limits_{{{R}_{n}}\subseteq {{P}_{n}}\left( C \right)}{\vartheta \left( {{R}_{n}}\cup \left\{ {{S}_{n}} \right\} \right)-\vartheta \left( {{R}_{n}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}} \\\end{align} \right)T$

For convenience,

let ${{\beta }_{j}}=\frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}},$ it can be found that the CIPFHGA operator is three phases:

• To obtain the $n \cdot \alpha_j \cdot \beta_i$, it multiplies the importance degrees of the PFNs $\alpha_j$ by the appropriate FM $\vartheta$ as well as multiply these PFNs by a balancing coefficient $n$.
• It orders every $n \cdot \alpha_j \cdot \beta_j(j=1,2, \ldots, n)$ in descending $\operatorname{order}\left(\tilde{\alpha}_{(1)}, \widetilde{\alpha}_{(2)}, \cdots, \tilde{\alpha}_{(n)}\right)$, where $\tilde{\alpha}_{(j)}$ is the $j_m^{\text {th }}$ greatest of $n \cdot \alpha_j \cdot \beta_j$.
• It gives each of these ordered PFNs $\tilde{\alpha}_{(j)}$ a weight $w_j$ $(j=1,2, \ldots, n)$ and then adds aggregated of the weighted PFNs $\tilde{\alpha}_{(j)}$ into a single one.

Theorem 6: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$ and has an weight vector $w=\left(w_1\right.$, $\left.w_2, \ldots, w_n\right)^T \quad$ with $\quad 0 \leq w_j \leq 1 \quad$ and $\quad \sum_{j=1}^n w_j=1, \quad \tilde{\alpha}_{(j)}=$ $\left\langle\mu_{\widetilde{\alpha}_{(j)}}, \eta_{\widetilde{\alpha}_{(j)}}, v_{\widetilde{\alpha}_{(j)}}\right\rangle$ be the $j^{\text {th }}$ largest of $\tilde{\alpha}_j$,

${{\tilde{\alpha }}_{j}}=n\cdot {{\alpha }_{j}}\cdot \frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)-\vartheta \left( {{R}_{j}} \right)}}},$

Then, using the CIPFHGA operator, their aggregated value is also a PFN

$CIPFHGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\left\langle \begin{align}  & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}}+{{\eta }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}}-}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}},1-\prod\limits_{j=1}^{n}{{{(1-{{\nu }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}}}} \\\end{align} \right\rangle \,\,$

Theorem 7: (Idempotency) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1$, $2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$ and has an weight vector $w=\left(w_1, w_2, \ldots, w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$. If all $\alpha_j(j=1,2, \ldots, n)$ are equal, i.e., $\alpha_j=$ $\alpha: \forall j$, then $\operatorname{CIPFHGA}\left(\alpha_1, \alpha_2, \cdots, \alpha_n\right)=\alpha .$

Theorem 8: (Monotonicity) Let $\alpha_{1 j}=\left(\mu_{\alpha_{1 j}}, \eta_{\alpha_{1 j}}, v_{\alpha_{1 j}}\right)$ and $\alpha_{2 j}=\left(\mu_{\alpha_{2 j}}, \eta_{\alpha_{2 j}}, v_{\alpha_{2 j}}\right)(j=1,2, \ldots, n)$ be a family of PFNs and $\vartheta$ be a FM on $S$ and has an weight vector $w=\left(w_1, w_2, \ldots\right.$, $\left.w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$. if $\mu_{\alpha_{1 j}} \leq \mu_{\alpha_{2 j}}, \eta_{\alpha_{1 j}} \geq$ $\eta_{\alpha_{2 j}}$ and $v_{\alpha_{1 j}} \geq v_{\alpha_{2 j}}$, then $\operatorname{CIPFHGA}\left(\alpha_{11}, \alpha_{12}, \cdots, \alpha_{1 n}\right) \leq$ $\operatorname{CIPFHGA}\left(\alpha_{21}, \alpha_{22}, \cdots, \alpha_{2 n}\right)$.

Theorem 9: (Boundedness) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=$ $1,2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$ and has a weight" vector $w=\left(w_1, w_2, \ldots, w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$.

If $\alpha^{-}=\left\langle\min _j\left(\mu_{\alpha_j}\right), \max _j\left(\eta_{\alpha_j}\right), \max _j\left(v_{\alpha_j}\right)\right\rangle$ and $\alpha^{+}=$ $\left\langle\max _j\left(\mu_{\alpha_j}\right), \min _j\left(\eta_{\alpha_j}\right), \min _j\left(v_{\alpha_j}\right)\right\rangle$, then $\alpha \leq \operatorname{CIPFHAA}\left(\alpha_1\right.$, $\left.\alpha_2, \cdots, \alpha_n\right) \leq \alpha^{+} .$.

Theorem 10: (Permutation) let $\alpha_j=\left(\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right) \alpha_{(j)}=$ $\left(\mu_{\alpha_{(j)}}, \eta_{\alpha_{(j)}}, v_{\alpha_{(j)}}\right)(j=1,2, \ldots, n)$ be two familys of PFNs on $S$, $\vartheta$ be a fuzzy measure on $S$ and has an weight vector $w=\left(w_1\right.$, $\left.w_2, \ldots, w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$. where $(j)$ denotes an arbitrary permutation of $j$. Then 

$\operatorname{CIPFHGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=\operatorname{CIPFHGA}\left(\alpha_{(1)}, \alpha_{(2)}, \ldots, \alpha_{(n)}\right)$.

Theorem 2 - Theorem 5 proofs are similar to Theorem 7 - Theorem 10 proofs, which are ignored here.

Example 2: Suppose w=(0.2429, 0.5142, 0.2429)^T by referring to (Xu [26]) then, the example 1, by using the CIPFHGA operator we have:

$\begin{align}  & \alpha _{1}^{{{\beta }_{1}}}=0.2298\cdot \left\langle 0.4,0.2,0.3 \right\rangle =\left\langle 0.1984,0.6908,0.0787 \right\rangle  \\ & \alpha _{2}^{{{\beta }_{2}}}=0.3333\cdot \left\langle 0.3,0.1,0.5 \right\rangle =\left\langle 0.2726,0.4642,0.2063 \right\rangle  \\ & \alpha _{3}^{{{\beta }_{3}}}=0.4369\cdot \left\langle 0.6,0.2,0.1 \right\rangle =\left\langle 0.4121,0.4950,0.0449 \right\rangle . \\\end{align}$

Then, $\quad S\left(\alpha_1^{\beta_1}\right)=-0.5711 \quad, \quad S\left(\alpha_2^{\beta_2}\right)=-0.3989$ and $S\left(\alpha_3^{\beta_3}\right)=-0.1278$.

Following the rules of section 2.2, we may conclude that based on these comparative laws,

${{\tilde{\alpha }}_{1}}=\alpha _{3}^{3{{\beta }_{3}}}=\left\langle 0.3734,0.3297,0.2180 \right\rangle$

${{\tilde{\alpha }}_{2}}=\alpha _{2}^{3{{\beta }_{2}}}=\left\langle 0.2999,0.1207,0.5000 \right\rangle$

${{\tilde{\alpha }}_{3}}=\alpha _{1}^{3{{\beta }_{1}}}=\left\langle 0.6251,0.1213,0.1287 \right\rangle .$

then,

$\begin{align}  & CIPFHGA\left( {{\alpha }_{1}},{{\alpha }_{2}},\cdots ,{{\alpha }_{n}} \right) \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.2429\cdot \left\langle 0.3734,0.3297,0.2180 \right\rangle \otimes  \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.5142\cdot \left\langle 0.2999,0.1207,0.5000 \right\rangle \otimes  \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.2429\cdot \left\langle 0.6251,0.1213,0.1287 \right\rangle \, \\\end{align}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left\langle 0.1556,0.1543,0.3621 \right\rangle .$

In this part, we will use the numerical example below to demonstrate the proposed approach. Assume that decision makers must select between four alternatives A₁, A₂, A₃ and A₄. Four attributes S₁, S₂, S₃ and S₄ where the attributes are cost attributes. The PF ratings of alternatives {A_i}i=1, 2, ..., m according to attributes{S_j}j=1, 2, ..., n are evaluated by decision makers and form the decision matrix is given below and the weight information of attribute w=(0.1550, 0.3450, 0.3450, 0.1550)^T by referring to (Xu [26]).

$\begin{align}  & \begin{matrix}   \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{1}} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{2}} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{3}} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{4}}  \\\end{matrix} \\ & \begin{matrix}   {{S}_{1}}  \\   {{S}_{2}}  \\   {{S}_{3}}  \\   {{S}_{4}}  \\\end{matrix}\left[ \begin{matrix}   \left\langle 0.5,0.2,0.3 \right\rangle  & \left\langle 0.2,0.3,0.1 \right\rangle  & \left\langle 0.2,0.3,0.3 \right\rangle  & \left\langle 0.7,0.2,0.1 \right\rangle   \\   \left\langle 0.4,0.3,0.2 \right\rangle  & \left\langle 0.6,0.2,0.1 \right\rangle  & \left\langle 0.4,0.3,0.3 \right\rangle  & \left\langle 0.5,0.2,0.3 \right\rangle   \\   \left\langle 0.6,0.1,0.1 \right\rangle  & \left\langle 0.4,0.2,0.3 \right\rangle  & \left\langle 0.5,0.2,0.3 \right\rangle  & \left\langle 0.3,0.4,0.1 \right\rangle   \\   \left\langle 0.5,0.3,0.1 \right\rangle  & \left\langle 0.7,0.1,0.1 \right\rangle  & \left\langle 0.4,0.3,0.2 \right\rangle  & \left\langle 0.6,0.1,0.2 \right\rangle   \\\end{matrix} \right] \\\end{align}$

Step 1: There is no need to convert the criteria because they all cost.

Step 2: Evaluate the FM on S_j. The FM are sets as follows:

$\vartheta \left( {{S}_{1}} \right)=0.23,\,\vartheta \left( {{S}_{2}} \right)=0.31,\,\vartheta \left( {{S}_{3}} \right)=0.18,$

$\vartheta \left( {{S}_{4}} \right)=0.24,\,\vartheta \left( {{S}_{1}},{{S}_{2}} \right)=0.5482,$

$\vartheta \left( {{S}_{1}},{{S}_{3}} \right)=0.4148,\,\vartheta \left( {{S}_{1}},{{S}_{4}} \right)=0.4764,$

$\vartheta \left( {{S}_{2}},{{S}_{3}} \right)=0.4964,\,\vartheta \left( {{S}_{2}},{{S}_{4}} \right)=0.5586,$

$\vartheta \left( {{S}_{3}},{{S}_{4}} \right)=0.4249,\,\vartheta \left( {{S}_{1}},{{S}_{2}},{{S}_{3}} \right)=0.7396,$

$\vartheta \left( {{S}_{1}},{{S}_{2}},{{S}_{4}} \right)=0.8033,\,\vartheta \left( {{S}_{1}},{{S}_{3}},{{S}_{4}} \right)=0.6663,$

$\vartheta \left( {{S}_{2}},{{S}_{3}},{{S}_{4}} \right)=0.7501,\,\vartheta \left( {{S}_{1}},{{S}_{2}},{{S}_{3}},{{S}_{4}} \right)=1.$

Step 3: Confirm the weighting vector w=(0.1550, 0.3450, 0.3450, 0.1550)^T by referring to (Xu [26]).

Step 4: Calculate the evaluations. As an example, consider the calculation of A₁.

Since ${{\left\langle 0.5,0.2,0.3 \right\rangle }^{0.2468}}=\left\langle 0.2435,0.6722,0.0843 \right\rangle ,$ ${{\left\langle 0.2,0.3,0.1 \right\rangle }^{0.3114}}=\left\langle 0.1185,0.6873,0.0823 \right\rangle ,$${{\left\langle 0.2,0.3,0.3 \right\rangle }^{0.1938}}=\left\langle 0.0824,0.7919,0.0663 \right\rangle $ and ${{\left\langle 0.7,0.2,0.1 \right\rangle }^{0.2480}}=\left\langle 0.3033,0.6709,0.0258 \right\rangle$

Their respective score values are -0.523, -0.6511, -0.7763, and -0.3934. We get their order as:

$\begin{align}  & \left\langle 0.3033,0.6709,0.0258 \right\rangle >\left\langle 0.2435,0.6722,0.0843 \right\rangle > \\ & \,\,\,\,\,\left\langle 0.1185,0.6873,0.0823 \right\rangle >\left\langle 0.0824,0.7919,0.0668 \right\rangle . \\\end{align}$

Hence, we have:

$\begin{align}  & CIPFHGA\left( {{\alpha }_{11}},{{\alpha }_{12}},{{\alpha }_{13}},{{\alpha }_{14}} \right)= \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\left\langle 0.3033,0.6709,0.0258 \right\rangle }^{4\cdot 0.1550}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\otimes {{\left\langle 0.2435,0.6722,0.0843 \right\rangle }^{4\cdot 0.3450}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\otimes {{\left\langle 0.1185,0.6873,0.0823 \right\rangle }^{4\cdot 0.3450}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\otimes {{\left\langle 0.0824,0.7919,0.0668 \right\rangle }^{4\cdot 0.1550}} \\\end{align}$

$\left\langle \begin{array}{c}(0.3033+0.6709)^{0.62}-0.6709^{0.62} \\ \cdot(0.2435+0.6722)^{1.38}-0.6722^{1.38}\\ \cdot(0.1185+0.6873)^{1.38}-0.6722^{1.38} \\ \cdot(0.0824+0.7919)^{0.62}-0.7919^{0.62},0.6709^{0.62} \\ \cdot 0.6722^{1.38} \cdot 0.6722^{1.38} \cdot 0.7919^{0.62}, \\ 1-\left[(1-0.0258)^{0.62} \cdot(1-0.0843)^{1.38} \cdot(1-0.0823)^{1.38} \cdot(1-0.0668)^{0.62}\right] \end{array} \right\rangle$

$=\langle 0.0134,0.0252,0.6839\rangle$

At the same reason, A₂ = ⟨0.0016, 0.2426, 0.2300⟩, A₃ = ⟨0.0015, 0.1988, 0.2449⟩ and A₄=⟨0.0033, 0.1831, 0.1473⟩.

Step 5: Rank the alternatives. The A_i(i =1, 2, 3, 4) score values are -0.6957, -0.4710, -0.4422 and -0.3271 respectively. Hence alternative can be ordered as A₄>A₃>A₂>A₁ the best alternative A₄.